Number Theory

Divisibility and PrimesDefinition. If a and b are integers and there is

some integer c such that a = b · c, then we say

that b divides a or is a factor or divisor of a and

write b|a.Definition (Prime Number).A prime number

is an integer greater than 1 whose only positive

divisors are itself and 1. A non-prime number

greater than 1 is called a composite number.

Theorem (The Fundamental Theorem of

Arithmetic).Every positive integer greater than

1 may be expressed as a product of primes and

this representation is unique up to the order in

which the factors are written.

Theorem.There are infinitely many prime num-

bers.

Proof. Suppose otherwise. Then there would

be a finite number n of primes, which we may

denote by p1, p2, p3, . . . , pn. Consider x = p1 ·p2 ·p3 . . . pn + 1. There must be some prime num-

ber greater than 1 which divides x, but clearly

x is not divisible by any of p1, p2, p3, . . . , pn.

This contradicts the assumption that there is

are finitely many primes, proving there are in-

finitely many.

Sieve of Erastothenes

The Sieve of Erastothenes is a technique which

may be used to determine all the prime num-

bers up to a certain size. One writes down all

the integers up to that size. One then crosses

out all the multiples of 2 (the even numbers)

greater than 2. At each step, one takes the

smallest number left whose multiples haven’t

been crossed out and crosses out all its multi-

ples. One is ultimately left only with the prime

numbers.

Test for Primality

One may check every integer less than the

number’s square root. If none are divisors,

then the integer is prime.

This may be seen by recognizing that if an

integer n is not prime, there must be integers

p ≤ q both dividing n. But then p2 ≤ pq ≤ n,

so p ≤√

n. So every non-prime number must

have a divisor no greater than its square root.

There are much more sophisticated tests for

primality.

Goldbach’s Conjecture

Goldbach’s Conjecture is that every even inte-

ger greater than 4 may be written as a sum of

two odd primes.

Goldbach’s Conjecture has been shown to hold

for all even integers up to 400 trillion, but has

not yet been proven in general. Hence, it re-

mains a conjecture rather than a theorem*.

Theorem (The Division Algorithm). If a, b

are integers with b > 0, then there exist unique

integers q, r such that a = q · b+ r with 0 ≤ r <

b. q is called the quotient and r is called the

remainder.

Note: The Division Algorithm is not an algo-

rithm!

Note: Any number which divides both a and b

also divides both b and r and visa versa.

Definition (Greatest Common Divisor).The

greatest common divisor of integers a and b is

the largest positive integer which divides both

a and b. We denote the greatest common di-

visor by gcd(a, b) or simply (a, b).

The Euclidean Algorithm gives a method (an

algorithm!) for finding the greatest common

divisor of any two positive integers:

Given a, b, we apply the Euclidean Algorithm

and find (a, b) = (b, r). We then apply the

Euclidean Algorithm to the pair b, r. We keep

repeating the process, each time getting a new

pair of numbers with the same gcd as a, b, until

we get two numbers such that one divides the

other. That divisor is the gcd we’re looking

for.

Modular ArithmeticDefinition (mod). If a is an integer and n is apositive integer, then a mod n is the remain-der obtained when we divide a by n using theEuclidean Algorithm.Definition (congruence). If n is a positive in-teger, two integers a, b are said to be congruentmodulo n if they both have the same remainderwhen divided by n. We write a ≡ b mod n.Corollary. a ≡ b mod n if and only if n|(a− b).

Modular arithmetic has many of the same prop-erties as ordinary arithmetic. We may defineaddition, subtraction and multiplication mod-ulo n because it is easily seen that if a ≡ b

mod n and c ≡ d mod n, then:

1. a + c ≡ b + d mod n

2. a− c ≡ b− d mod n

3. a · b ≡ b · d mod n

Divisibility Tests

Modular arithmetic may be used to show the

validity of a number of common divisibility tests.

Casting Out Nines

A test for divisibility is called Casting Out Nines:

Theorem. A positive integer is divisible by 9 if

and only if the sum of its digits is divisible by

9.

Proof. Since 10 ≡ 1 mod 9, it follows that

10n ≡ 1 mod 9 for any positive integer n. Given

any integer N , we may write N = am · 10m +

am−1 ·10m−1+am−2 ·10m−2+ . . . a0 ·100, where

a0, a1, a2, . . . am are the digits in N . But then

N ≡ am · 1 + am−1 · 1 + · · ·+ a0 mod 9.

Essentially the same reasoning shows:

Theorem. A positive integer is divisible by 3 if

and only if the sum of its digits is divisible by

3.

A variation gives a method called Casting out

Elevens for testing divisibility by 11. It’s based

on the fact that 10 ≡ −1 mod 11, so 10n ≡(−1)n mod 11.

Theorem (Casting Out Elevens). A positive

integer is divisible by 11 if and only if the al-

ternating sum of its digits is divisible by 11.

Proof. Since 10 ≡ −1 mod 9, it follows that

10n ≡ (−1)n mod 11 for any positive integer

n. Given any integer N , we may write N = am ·10m+am−1 ·10m−1+am−2 ·10m−2+ . . . a0 ·100,

where a0, a1, a2, . . . am are the digits in N . But

then N ≡ am ·(−1)m+am−1 ·(−1)m−1+ · · ·+a0

mod 11 ≡ a0 − a1 + a2 − a3 + · · · + (−1)mam

mod 11.

Other Tests

• Divisibility By 2 – The units digit must be

even.

• Divisibility By 4 – The number formed by

its last two digits must be divisible by 4.

• Divisibility By 5 – The units digit must be

0 or 5.

• Divisibility By 6 – It must be even and di-

visible by 3.

• Divisibility By 7 – When the units digit is

doubled and subtracted from the number

formed by the remaining digits, the result-

ing number must be divisible by 7. (To ver-

ify, write the original number in the form

10a + b ≡ 3a + b mod 7, so the resulting

number is a − 2b, and check the possible

ways for 3a + b to be divisible by 7.)

• Divisibility By 8 – The number formed by

its last three digits must be divisible by 8.

• Divisibility By 10 – Its last digit must be 0.

Check DigitsDefinition (Check Digit). A check digit is

an extra digit tacked onto a number which

is mathematically related in some way to the

other digits.

Example: Airline Tickets – The check digit is

the main part mod 7.

Example: U.S. Postal Service Money Orders –

The check digit is the main part mod 9.

These do not catch all single-digit errors nor

do they catch transposition errors.

Bank Identification Number Check Digit For-

mula: Every bank has a nine digit identification

number of the form a8a7a6a5a4a3a2a1a0 where

a0 = (7a8+3a7+9a6+7a5+3a4+9a3+7a2+

3a1) mod 10.

UPC Number Check Digit Formula: a0 is cho-

sen so (3a11+a10+3a9+a8+3a7+a6+3a5+

a4 + 3a3 + a2 + 3a1) ≡ 0 mod 10.

ISBN Check Digit Formula: a0 ≡ (a9 + 2a8 +

3a7+4a6+5a5+6a4+7a3+8a2+9a1) mod 11.

There is a check digit method that detects all

single-digit and transposition errors and only

generates 0 through 9 as a check digit.

Tournament Scheduling

Problem: How do we schedule the teams play-

ing in a round-robin tournament?

Solution:

Let N be the number of teams in the tourna-

ment and number the teams 1,2,3, . . . , N .

Let Tm,r be the team which Team m plays in

Round r.

If there is an odd number of teams, we let Tm,r

be the unique integer between 1 and N such

that

Tm,r ≡ r −m mod N.

If Tm,r = m, then Team m gets a bye.

If there is an even number of teams, we sched-

ule the teams as if there was one fewer team

and let the team that would otherwise get a

bye play the last team.

CryptologyDefinition (Cryptology).Cryptology is the dis-cipline of encoding and decoding messages.

Cryptology is critical in everyday life today.Our banking system, including the ability touse ATM’s and to do online banking, wouldcollapse without the ability to securely trans-mit financial information over public networks.

Cryptology has played a crucial role in history.Many believe that World War II was shortenedby several years because the Allies were able tocrack the secret codes used by the Axis powers.Definition (Cipher). A cipher is a method forencoding messages.Definition (Plaintext). Plaintext refers to theoriginal text that is being encoded.Definition (Ciphertext). Ciphertext refers tothe encoded message.Definition (Enciphering, Encryption). Theprocess of encoding a message is sometimesreferred to as enciphering or encryption.

Definition (Deciphering, Decryption). The

process of Decoding a message is sometimes

referred to as deciphering or decryption.

The Caesar Cipher

The Caesar Cipher is one of the earliest known

ciphers and was used by Julius Casar. Each let-

ter in a message is simply replaced by the letter

coming three letters after it in the alphabet.

Obvious problem: What about x, y and z?

Obvious solution: Replace them with a, b and

c.

We may make this somewhat quantitative by

assigning a numerical value to each letter: 0

to A, 1 to B, 2 to C, . . . , 25 to Z. If we let P

represent the numerical value of a given letter

in plaintext and C represent the number it is

replaced by in ciphertext, we have

C ≡ (P + 3) mod 26.

To decode, we have

P ≡ (C − 3) mod 26.

More generally, we might use a shift other than

3 and let C ≡ (P + b) mod 26 for some other

integer b.

We might mix things up a little more and let

C ≡ (aP + b) mod 26 (1)

for some choice of integers a and b. Such a

cipher would be called an affine cipher.

To decode, we could try to solve the congru-

ence (1) for P in terms of C. We might pro-

ceed as follows:

C ≡ aP + b mod 26,

aP ≡ C − b mod 26,

P ≡ a−1(C − b) mod 26.

Question: What is a−1, the multiplicative in-

verse of a?

It would have to be a number such that a·a−1 ≡1 mod 26.

It turns out that not all integers have such an

inverse. For example, no even numbers could

have inverses mod 26 since any multiple of

an even number would be even and could only

be congruent to another even integer and thus

could not possibly be congruent to 1.

Theorem 1. An integer a has a multiplicative

inverse mod n if and only if a and n have

no factors in common other than 1, in other

words, if (a, n) = 1.

The Hill Cipher

With the Hill Cipher, blocks of letters are en-

coded simultaneously rather than encoding let-

ters separately in a method similar to an affine

cipher. For a block of two letters, P1, P2, the

corresponding encoded letters C1, C2 would be

determined by the formulas

C1 ≡ (aP1 + bP2) mod 26

C2 ≡ (cP1 + dP2) mod 26,

where a, b, c, d are integers. It turns out we will

need (ad − bc,26) = 1 in order to be able to

decipher the encoded message. To see this,

we may try to solve for P1 and P2 in terms of

C1 and C2 the same way we would if we were

dealing with ordinary equations–by matching

coefficients and using elimination.

We might match the coefficients of P2 by mul-

tiplying the first equation by d and the second

by b to get

dC1 ≡ (adP1 + bdP2) mod 26

bC2 ≡ (bcP1 + bdP2) mod 26,

.

Subtracting, we get

dC1 − bC2 ≡ (ad− bc)P1 mod 26.

We need ad − bc to have an inverse mod 26

in order to solve:

P1 ≡ (ad− bc)−1(dC1 − bC2) mod 26.

Similarly, we may find

P2 ≡ (ad− bc)−1(aC2 − cC1) mod 26.

Variations may be used for larger blocks of let-

ters.

The RSA Public Key System

In public key systems, the encoding method

is made public, so that anyone may send an

encoded message, but the decoding method

is known only to the recipient. This depends

on the difficulty of determining the decoding

method even if the encoding method is known.

The RSA Public Key System was created by

Ron Rivest, Adi Shamir and Len Adelman in

1975. It is probably the most widely used sys-

tem today.

The Method

Two large primes, p, q, are determined along

with their product n = pq and a positive integer

r relatively prime to both p− 1 and q− 1. The

values of n and r are published. A block of

letters is then encoded by letting

C ≡ P r mod n.

To decode the message, the recipient deter-

mines the multiplicative inverse k = r−1 mod (p−1)(q − 1) and calculates

P ≡ Ck mod n.